Cell decomposition and classification of definable sets in p-optimal fields

TL;DR

This paper establishes cell decomposition, definable Skolem functions, and strong p-minimality for p-optimal fields, leading to a classification of definable sets and functions in these fields, especially those satisfying the Extreme Value Property.

Contribution

It proves cell decomposition and related properties for p-optimal fields, extending Denef's methods, and develops a preparation theorem for definable functions in these fields.

Findings

Cell decomposition holds in p-optimal fields.

Definable Skolem functions exist in these fields.

Infinite definable sets are classified by dimension.

Abstract

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef's paper [Invent. Math, 77 (1984)]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-optimal fields satisfying the Extreme Value Property (a property which in particular holds in fields which are elementarily equivalent to a p-adic one). For such fields K, we prove that every definable subset of KxK^d whose fibers are inverse images by the valuation of subsets of the value group, are semi-algebraic. Combining the two we get a preparation theorem for definable functions on p-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are isomorphic iff they have the same dimension.